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Given a subgroup H of a group (G , +) , a (G , H , k , 1) difference family (DF) is a set F of k -subsets of G such that { f − f ′ : f , f ′ ∈ F , f ≠ f ′ , F ∈ F } = G ∖ H. Let g Z g h be the subgroup of order h in Z g h generated by g. A (Z g h , g Z g h , k , 1) -DF is called cyclic and written as a (g h , h , k , 1) -CDF. This paper shows that for h ∈ { 2 , 3 , 6 } , there exists a (g h , h , 4 , 1) -CDF if and only if g h ≡ h (mod 12) , g ⩾ 4 and (g , h) ∉ { (9 , 3) , (5 , 6) }. As a corollary, it is shown that a 1-rotational Steiner system S (2 , 4 , v) exists if and only if v ≡ 4 (mod 12) and v ≠ 28. This solves the long-standing open problem on the existence of a 1-rotational S (2 , 4 , v). As another corollary, we establish the existence of an optimal (v , 4 , 1) -optical orthogonal code with ⌊ (v − 1) / 12 ⌋ codewords for any positive integer v ≡ 1 , 2 , 3 , 4 , 6 (mod 12) and v ≠ 25. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic 3-ary constant-weight codes with weight four and minimum distance six. [ABSTRACT FROM AUTHOR]

If S is a transitive metric space, then | C | ⋅ | A | ≤ | S | for any distance- d code C and a set A , "anticode", of diameter less than d. For every Steiner S (t , k , n) system S , we show the existence of a q -ary constant-weight code C of length n , weight k (or n − k), and distance d = 2 k − t + 1 (respectively, d = n − t + 1) and an anticode A of diameter d − 1 such that the pair (C , A) attains the code–anticode bound and the supports of the codewords of C are the blocks of S (respectively, the complements of the blocks of S). We study the problem of estimating the minimum value of q for which such a code exists, and find that minimum for small values of t. [ABSTRACT FROM AUTHOR]

The research on orthogonal Steiner systems S (t , k , v) was initiated in 1968. For (t , k) ∈ { (2 , 3) , (3 , 4) } , this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t ≥ 3 and k ≥ 5 , only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S (3 , 5 , v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S (3 , 5 , 4 m + 1) is established for any even m ≥ 2 ; additionally a pair of orthogonal G-designs G ( 4 m + 1 5 , 5 , 5 , 3) is displayed for any odd m ≥ 3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S (5 , 8 , 24) are shown to exist. [ABSTRACT FROM AUTHOR]

We construct a Ramsey class whose objects are Steiner systems. In contrast to the situation with general r -uniform hypergraphs, it turns out that simply putting linear orders on their sets of vertices is not enough for this purpose: one also has to strengthen the notion of subobjects used from “induced subsystems” to something we call “strongly induced subsystems”. Moreover we study the Ramsey properties of other classes of Steiner systems obtained from this class by either forgetting the order or by working with the usual notion of subsystems. This leads to a perhaps surprising induced Ramsey theorem in which designs get coloured. [ABSTRACT FROM AUTHOR]

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph. [ABSTRACT FROM AUTHOR]

The family Σ r consists of all r -graphs with three edges D 1 , D 2 , D 3 such that | D 1 ∩ D 2 | = r − 1 and D 1 △ D 2 ⊆ D 3 . A generalized triangle , T r ∈ Σ r is an r -graph on { 1 , 2 , … , 2 r − 1 } with three edges D 1 , D 2 , D 3 , such that D 1 = { 1 , 2 , … , r − 1 , r } , D 2 = { 1 , 2 , … , r − 1 , r + 1 } and D 3 = { r , r + 1 , … , 2 r − 1 } . Frankl and Füredi conjectured that for all r ≥ 4 , ex ( n , Σ r ) = ex ( n , T r ) for all sufficiently large n and they also proved it for r = 3 . Later, Pikhurko showed that the conjecture holds for r = 4 . In this paper we determine ex ( n , T 5 ) and ex ( n , T 6 ) for sufficiently large n , proving the conjecture for r = 5 , 6 . [ABSTRACT FROM AUTHOR]

Abstract: For each positive integer n, we construct a Steiner triple system of order with no almost parallel class; that is, with no set of disjoint triples. In fact, we construct families of -designs with an analogous property. The only previously known examples of Steiner triple systems of order congruent to without almost parallel classes were the projective triple systems of order with n odd, and 2 of the 11,084,874,829 Steiner triple systems of order 19. [Copyright &y& Elsevier]

Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v ≡ 1 (mod 6) , it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic (v , k , λ) -designs with 1 ⩽ λ ⩽ k − 1. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all v , but we conjecture that it holds whenever v is sufficiently large compared to k. We confirm that the generalization of the conjecture holds when v is prime and λ = 1 and also when λ ⩽ (k − 1) / 2 and v is sufficiently large compared to k. As a corollary, we show that for any k ⩾ 3 , with the possible exception of finitely many composite orders v , every cyclic (v , k , 1) -design without short orbits is generated by a (v , k , 1) -disjoint difference family. [ABSTRACT FROM AUTHOR]

Abstract: The block graph of a Steiner triple system of order v is a strongly regular graph. For large v, every strongly regular graph with these parameters is the block graph of a Steiner triple system, but exceptions exist for small orders. An explanation for some of the exceptional graphs is here provided via the concept of switching. (Group divisible designs corresponding to) Latin squares are also treated in an analogous way. Many new strongly regular graphs are obtained by switching and by constructing graphs with prescribed automorphisms. In particular, new strongly regular graphs with the following parameters that do not come from Steiner triple systems or Latin squares are found: , , , , , and . [Copyright &y& Elsevier]

Abstract: It has been conjectured that any partial triple system of order u and index λ can be embedded in a triple system of order v and index λ whenever , is even and . This conjecture is known to hold for and for all even . Here the conjecture is proven for all remaining values of λ when . [Copyright &y& Elsevier]

Abstract: We give a construction of a 2- design starting from a Steiner system and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs. [ABSTRACT FROM AUTHOR]

Abstract: This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron–Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6- designs with , except possibly when the group is with or 3, and e is an odd prime power. [Copyright &y& Elsevier]

Abstract: We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907. [Copyright &y& Elsevier]

Abstract: The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is . Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15—are tabulated. A consistency check based on double counting is carried out to gain confidence in the correctness of the classification. [Copyright &y& Elsevier]

Abstract: In [Z. Füredi, Turán type problems, in: Surveys in Combinatorics, Guildford, 1991, in: London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 253–300, MR1161467 (93d:05072)], Füredi raised a conjecture about the maximum size of L-intersecting families. In this note, we address a variant of this conjecture. In particular, we show that for any Steiner triple system on , there exist a family of k-sets on with and such that for every the family is isomorphic to . [Copyright &y& Elsevier]

Abstract: A Steiner triple system of order (briefly STS) consists of a -element set X and a collection of 3-element subsets of X, called blocks, such that every pair of distinct points in X is contained in a unique block. A large set of disjoint STS (briefly LSTS) is a partition of all 3-subsets (triples) of X into STS. In 1983–1984, Lu Jiaxi first proved that there exists an LSTS for any or with six possible exceptions and a definite exception . In 1989, Teirlinck solved the existence of LSTS for the remaining six orders. Since their proof is very complicated, it is much desired to find a simple proof. For this purpose, we give a new proof which is mainly based on the 3-wise balanced designs and partitionable candelabra systems. [Copyright &y& Elsevier]

Any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if v≡1,3 (mod 6) and v⩾3u−2. [Copyright &y& Elsevier]

Orthogonal quadruple systems are defined and investigated. Orthogonal quadruple systems with an additional nesting property are shown to provide a new construction of 3-frames F(3,4,n{2}) (Design Codes Cryptogr. 4 (1994) 5). Constructions for pairs of nested orthogonal quadruple systems are provided. Applications to 2-resolvable Steiner quadruple systems and highly nonlinear functions are discussed. [Copyright &y& Elsevier]

Two Steiner triple systems (STS) are orthogonal if their sets of triples are disjoint, and two disjoint pairs of points defining intersecting triples in one system fail to do so in the other. In 1994, it was shown (Canad. J. Math. 46(2) (1994) 239–252) that there exist a pair of orthogonal Steiner triple systems of order v for all v≡1,3 (mod 6), with v⩾7, v≠9. In this paper we show that there exist three pairwise orthogonal Steiner triple systems of order v for all v≡1 (mod 6), with v⩾19 and for all v≡3 (mod 6), with v⩾27 with only 24 possible exceptions. [Copyright &y& Elsevier]

A Steiner triple system (STS(v)) is said to be 3-balanced if every 3-colouring of it is equitable; that is, if the cardinalities of the colour classes differ by at most one. A 3-colouring, φ, of an STS(v) is unique if there is no other way of 3-colouring the STS(v) except possibly by permuting the colours of φ. We show that for every admissible v⩾25, there exists a 3-balanced STS(v) with a unique 3-colouring and an STS(v) which has a unique, non-equitable 3-colouring. [Copyright &y& Elsevier]

The Erdo˝s–Stone–Simonovits Theorem implies that the Tura´n density of a family of graphs is the minimum of the Tura´n densities of the individual graphs from the family. It was conjectured by Mubayi and Ro¨dl (J. Combin. Theory Ser. A, submitted) that this is not necessarily true for hypergraphs, in particular for triple systems. We give an example, which shows that their conjecture is true. [Copyright &y& Elsevier]

For a family of r-graphs F the Tura´n number ex (n, F) is the maximum number of edges in an n vertex r-graph that does not contain any member of F. The Tura´n density π(F)=limlower limit n→∞ ex(n,F)/((nr). When F is an r-graph, π(F)≠0, and r>2, determining π(F) is a notoriously hard problem, even for very simple r-graphs F. For example, when r=3, the value of π(F) is known for very few (<10) irreducible r-graphs. Building upon a method developed recently by de Caen and Fu¨redi (J. Combin. Theory Ser. B78 (2000), 274–276), we determine the Tura´n densities of several 3-graphs that were not previously known. Using this method, we also give a new proof of a result of Frankl and Fu¨redi (Combinatorica 3 (1983), 341–349) that π(H)=2/9, where H has edges 123,124,345. Let F(3,2) be the 3-graph 123,145,245,345, let K−4 be the 3-graph 123,124,234, and let C5 be the 3-graph 123,234,345,451,512. We prove

Perfect codes and optimal anticodes in the Grassman graph Gq(n, k) are examined. It is shown that the vertices of the Grassman graph cannot be partitioned into optimal anticodes, with a possible exception when n=2k. We further examine properties of diameter perfect codes in the graph. These codes are known to be similar to Steiner systems. We discuss the connection between these systems and “real” Steiner systems. [Copyright &y& Elsevier]

Let K be an abelian group of order v. A Steiner quadruple system of order v (SQS (v)) (K , B) is called symmetric K -invariant if for each B ∈ B , it holds that B + x ∈ B for each x ∈ K and B = − B + y for some y ∈ K. When the Sylow 2-subgroup of K is cyclic, a necessary and sufficient condition for the existence of a symmetric K -invariant SQS (v) was given by Munemasa and Sawa, which is a generalization of a necessary and sufficient condition for the existence of a symmetric cyclic SQS (v) shown in Piotrowski's thesis in 1985. In this paper, we prove that a symmetric K -invariant SQS (v) exists if and only if v ≡ 2 , 4 (mod 6) , the order of each element of K is not divisible by 8, and there exists a symmetric cyclic SQS (2 p) for any odd prime divisor p of v. [ABSTRACT FROM AUTHOR]

In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fraïssé limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fraïssé's construction and its variants are rich sources of examples that are central to model-theoretic classification theory, and recently infinite Steiner systems obtained via Fraïssé-type constructions have received attention from the model theory community. [ABSTRACT FROM AUTHOR]

Kramer and Magliveras constructed simple 5-(24,8,λ) designs for λ&les;9. Betten et al. constructed simple 5-(24,8,λ) designs for λ=16,17,…,484. In this paper, we construct 15 mutually disjoint Steiner systems S(5,8,24) and thus we construct simple 5-(24,8,λ) design for λ=10,11,12,13,14 and 15. As a consequence there exists a simple 5-(24,8,λ) design for any positive integer λ&les;λmax=969. [Copyright &y& Elsevier]